Vector Notation & derivation in Cylindrical Coordinates - Navier-Stokes equation
- Using, vector notation to write Navier-Stokes and continuity equations for incompressible flow we have
| (24.21) |
and
| (24.22) |
- we have four unknown quantities, u, v, w and p ,
- we also have four equations, - equations of motion in three directions and the continuity equation.
- In principle, these equations are solvable but to date generalized solution is not available due to the complex nature of the set of these equations.
- The highest order terms, which come from the viscous forces, are linear and of second order
- The first order convective terms are non-linear and hence, the set is termed as quasi-linear.
- Navier-Stokes equations in cylindrical coordinate (Fig. 24.2) are useful in solving many problems. If , and denote the velocity components along the radial, cross-radial and axial directions respectively, then for the case of incompressible flow, Eqs (24.21) and (24.22) lead to the following system of equations:
FIG 24.2 Cylindrical polar coordinate and the velocity components
| (24.23a) |
| (24.23b) |
| (24.23c) |
| (24.24)
|